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The angle of intersection of the curves $x^2-y^2=5$ and $\frac{x^2}{18}+\frac{y^2}{8}=1$ is:

Examine
the continuity of f,
where f is
defined by

The graph of sin x meets x axis at

If the acute angle between two lines is $\frac{\pi }{4}$ and slope of one of them is $\frac{1}{2}$. Find the slope of the other line.

Prove the following:
cot 4x ( sin 5x + sin 3x) = cot x (sin 5x- sin 3x)

Probability of getting a sum of $9$ on two throws of a die is

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the follwing is a valid choice for the ordered triplets $(a,b,c)$?

${\int }_0^{\pi /2}\log \left(\sin x\right)dx=$

(i) How many terms are there in the A.P. 7, 10, 13, ..... 43 ?

(ii) How many terms are there in the A.P. $-1,\frac{-5}{6}-\frac{2}{3},\frac{1}{2},\dots \dots \frac{10}{3}$ ?

Let k be an integer such that the triangle with vertices (k,-3k), (5, k) and (-k, 2) has area 28 sq units. Then, the orthocentre of this triangle is at the point

If $\frac{(1-3p)}{2}$, $\frac{(1+4p)}{3}$, $\frac{(1+p)}{6}$ are the probabilities of three mutually exclusive and exhaustive events, then the set of all value of p is

(i) Is 68 a term of the A.P. 7, 10, 13, ..... ?

(ii) Is 302 a term of the A.P. 3, 8, 13, ..... ?

Let $0\le x,y,z\le \frac{\pi }{2}$ be such that $\sin x\sin y\cos z=\frac{1}{2\sqrt{2}},$ $\frac{{\sin }^{2}x{\sin }^{3}y}{\cos z}=\frac{1}{4\sqrt{2}}$ and $\frac{\sin x{\sin }^{4}y}{{\cos }^{2}z}=\frac{1}{8}.$ Then which of the following is/are CORRECT?

Write the general solution of $ta{n}^{2}2x=1.$

If $f^{\prime }\left(\tan x\right)=\cos 2x+\sin 2x\forall x\in R-\left\{\frac{(2n+1)\pi }{2}\right\},n\in Z,$ then $f\left(x\right)$ can be

(where $C$ is constant of integration)

Given $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{b}=2\hat{i}+3\hat{j}+\hat{k},\overrightarrow{c}=8\hat{i}+13\hat{j}+9\hat{k},$ the linear relation among them if possible is

If $\underset{-\pi /4}{\overset{3\pi /4}{\int }}\frac{e^{\pi /4}dx}{(e^x+e^{\pi /4})(\sin x+\cos x)}=\lambda \underset{-\pi /2}{\overset{\pi /2}{\int }}\sec xdx,$ then $\lambda$ is equal to

If $tanA=\frac{1-cosB}{sinB}$, then find the value of tan 2A.

tan(x/2)/2 + tan(x/4)/4 + ....... + tan(x/2^n)/2^n = cot(x/2^n)/2^n - cotx by principle of mathematical induction

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^2-9y^2+32x+36y-164=0$, and its foci is

Area bounded by the curve y = log x, x - axis and the ordinates x = 1, x = 2 is [MP PET 2004]

Int dx/√2ax-x,^2 = a^n sin^-1[x/a-1] .

The value of n is
(a) 0 (b) —1
(c) 1 (d) none of these.
You may use dimensional analysis to solve the problem.

If the function g(x) is defined by $g\left(x\right)=\frac{x^{200}}{200}+\frac{x^{199}}{199}+\frac{x^{198}}{198}+\dots \dots ..+\frac{x^2}{2}+x+5,$ then $g^{\prime }\left(0\right)=\dots \dots \dots \dots .$

The nearest point on the circle $x^2+y^2-6x+4y-12=0$ from the point $P(-5,4)$ is $Q(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

$Ifx+\frac{1}{x}=5,then(x^3+\frac{1}{x^3})-5(x^2+\frac{1}{x^2})+(x+\frac{1}{x})$ is equal to

In a matrix A of order 3, find $\left(\begin{array}{c}3A\end{array}\right|$?

The set of real values of x for which $lo{g}_{0.2}\frac{x+2}{x}\le 1$ is

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100m$ long is supported by vertical wires attached to the cable, the longest wire being $30m$ and the shortest being 6 m. Find the length of a supporting wire attached to the roadway $18m$ from the middle.

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it.The resulting mixture is to be more than 4% but less than 6% boric acid.If there are 640 liters of the 8% solution,how many liters of 2% solution will have to be added ?